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53rd AIAA Aerospace Sciences Meeting, January 5 - 9 2015, Kissimmee, FL High-Order Moving Overlapping Grid Methodology for Aerospace Applications Brandon Merrill * , Yulia Peet The development of a three dimensional spectrally accurate moving overlapping grid technique that is utilized to solve for the flow around complex moving geometries opens doors to various practical applications that have previously been difficult to model. This method provides a straightforward approach to managing grid resolution near moving struc- tures without the need for mesh restructuring techniques. Building upon our previously validated overlapping mesh methodology, we have developed a moving overlapping mesh method in a high-order spectral element computational fluid dynamics solver. Current validation tests on the two dimensional moving overlapping grid methodology have shown spectral accuracy of the coupled solution for both translating and rotating meshes, and re- sults for fluid interactions with translating structures that correlate well with experimental and other computational data. A three dimensional simulation demonstrates robustness of the method and the ability to represent flow around complex structures moving in complex ways, paving the way to accurately model the flow through a rotating wind turbine. I. Introduction The ability to flexibly and efficiently model the interaction of fluid with moving structures allows for analysis of numerous problems that have previously been very difficult to simulate. The motion of a three dimensional mesh able to move with six degrees of freedom (translational and rotational) envelops the possibility to model a wide range of useful applications spanning from the flow around helicopter blades, wind turbines, propellers, to flows in engines, mixed tanks, stirred reactors, and much more. The use of moving overlapping grids, especially when one or both of the grids contain solid bodies, enables a straightforward approach to managing grid resolution near walls without the need for grid restructuring techniques. The earliest known technique in domain decomposition methods was developed by Schwarz in 1870 with the introduction of the Schwarz Alternating Method. Schwarz provided a foundation upon which almost all other domain decomposition methods are built. Additional fundamental work with overlapping grids was performed in the 1960’s by Volkov 1 who called his technique the Composite Mesh Difference Method that used finite difference methods on overlapping two dimensional grids. Since that time, many others have built upon his work improving the Composite Mesh Difference Method. 24 This work in composite mesh methods led to the development of techniques like those produced by Henshaw et al. 5 Henshaw’s work involves a moving overlapping grid technique in two dimensions which utilizes boundary fitted grids that move on an overlapped stationary background Cartesian grid where one grid determines data for adaptive mesh refinement. Their method uses a second order extension of Godunov’s method with adaptive mesh refinement for spatial discretization. Another derivation of multigrid methods is employed by Rivera et al. 6 who has developed a sliding mesh technique for finite element simulations. Their technique utilizes three dimensional meshes that remain adjacent to each other, without overlap, and one mesh is constrained to move. When using this method for rotational motion, the rotating mesh must necessarily be circular. Additional research has been performed by many others using sliding mesh techniques to simulate three dimensional flows with complex geometries. 79 In addition, there has been much work done with respect * PhD Candidate, School for Engineering of Matter, Transport, and Energy (SEMTE), Arizona State University; E-mail: [email protected] Assistant Professor, School for Engineering of Matter, Transport and Energy (SEMTE), Arizona State University Copyright c 2015 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. 1 of 9 American Institute of Aeronautics and Astronautics

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53rd AIAA Aerospace Sciences Meeting, January 5 - 9 2015, Kissimmee, FL

High-Order Moving Overlapping Grid Methodology

for Aerospace Applications

Brandon Merrill∗, Yulia Peet†

The development of a three dimensional spectrally accurate moving overlapping gridtechnique that is utilized to solve for the flow around complex moving geometries opensdoors to various practical applications that have previously been difficult to model. Thismethod provides a straightforward approach to managing grid resolution near moving struc-tures without the need for mesh restructuring techniques. Building upon our previouslyvalidated overlapping mesh methodology, we have developed a moving overlapping meshmethod in a high-order spectral element computational fluid dynamics solver. Currentvalidation tests on the two dimensional moving overlapping grid methodology have shownspectral accuracy of the coupled solution for both translating and rotating meshes, and re-sults for fluid interactions with translating structures that correlate well with experimentaland other computational data. A three dimensional simulation demonstrates robustness ofthe method and the ability to represent flow around complex structures moving in complexways, paving the way to accurately model the flow through a rotating wind turbine.

I. Introduction

The ability to flexibly and efficiently model the interaction of fluid with moving structures allows foranalysis of numerous problems that have previously been very difficult to simulate. The motion of a threedimensional mesh able to move with six degrees of freedom (translational and rotational) envelops thepossibility to model a wide range of useful applications spanning from the flow around helicopter blades, windturbines, propellers, to flows in engines, mixed tanks, stirred reactors, and much more. The use of movingoverlapping grids, especially when one or both of the grids contain solid bodies, enables a straightforwardapproach to managing grid resolution near walls without the need for grid restructuring techniques.

The earliest known technique in domain decomposition methods was developed by Schwarz in 1870 withthe introduction of the Schwarz Alternating Method. Schwarz provided a foundation upon which almost allother domain decomposition methods are built. Additional fundamental work with overlapping grids wasperformed in the 1960’s by Volkov1 who called his technique the Composite Mesh Difference Method thatused finite difference methods on overlapping two dimensional grids. Since that time, many others havebuilt upon his work improving the Composite Mesh Difference Method.2–4 This work in composite meshmethods led to the development of techniques like those produced by Henshaw et al.5 Henshaw’s workinvolves a moving overlapping grid technique in two dimensions which utilizes boundary fitted grids thatmove on an overlapped stationary background Cartesian grid where one grid determines data for adaptivemesh refinement. Their method uses a second order extension of Godunov’s method with adaptive meshrefinement for spatial discretization. Another derivation of multigrid methods is employed by Rivera etal.6 who has developed a sliding mesh technique for finite element simulations. Their technique utilizesthree dimensional meshes that remain adjacent to each other, without overlap, and one mesh is constrainedto move. When using this method for rotational motion, the rotating mesh must necessarily be circular.Additional research has been performed by many others using sliding mesh techniques to simulate threedimensional flows with complex geometries.7–9 In addition, there has been much work done with respect

∗PhD Candidate, School for Engineering of Matter, Transport, and Energy (SEMTE), Arizona State University; E-mail:[email protected]†Assistant Professor, School for Engineering of Matter, Transport and Energy (SEMTE), Arizona State UniversityCopyright c© 2015 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free

license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by thecopyright owner.

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to moving objects on a single mesh using numerous methods including the Arbitrary Lagrangian-Eulerianmethod.10–13 Our research will improve upon previous methods by using the Spectral Element Methodon three dimensional meshes, which maintains spectral accuracy for spatial discretization, and by utilizingmoving overlapping grid techniques with spectral interpolation for flexibility of motion, and computationalefficiency.

Our research builds upon a domain decomposition technique employed within the computational fluiddynamics (CFD) code, Nek5000, developed at the Mathematics and Computer Science Division of ArgonneNational Laboratory.14 The governing equations of incompressible fluid flow (the incompressible Navier-Stokes equations) will be expressed using the Eulerian formulation for stationary meshes, which requiresthe computational domain to be fixed in space, and for moving meshes, the Arbitrary Lagrangian-Eulerianformulation is used, which allows the velocity of the mesh to be arbitrarily specified. In all overlapping cases,the values at the subdomain interfaces are handled using spectral interpolation with an explicit extrapolationmethod.

Building upon our overlapping mesh methodology, work was done to enable the use of moving meshes.We used an Arbitrary Lagrangian-Eulerian formulation as a platform for handling non-stationarity of themesh. We have also performed simulations to test the stability and validity of the code. Convecting eddysimulations were used to verify the spatial and temporal accuracy of the method. The capacity to accuratelyrepresent interaction of fluid with the moving bodies was investigated using simulations representing a twodimensional oscillating cylinder in uniform flow. Robustness and flexibility of the method, in preparationfor modeling wind turbine flows, is demonstrated by a simulation of two rotating cylinders in a uniform flowfield.

In this paper, we present the details and results of work done in the development and validation of ourmoving overlapping grid methodology. Section II will give an overview of the techniques and formulationsused to develop the moving overlapping mesh methods. Section III presents the current work done invalidating our moving overlapping mesh method and Section IV will conclude with a brief summary and anoverview of intentions for future work.

II. Methods

Our moving overlapping mesh method is employed within the Nek5000 computational fluid dynamicsframework14 which utilizes the Spectral Element Method for spatial discretization, and allows for up to thirdorder temporal discretization for solutions to the incompressible Navier-Stokes equations 1. It improves uponour stationary overlapping mesh methodology by allowing for rotation and/or translation of one of the grids.

∂u∂t + u · ∇u = −∇p + 1

Re∇2u

∇ · u = 0 (1)

II.A. Handling Interfaces Between Meshes

Values at mesh interfaces are determined using spectral interpolation on values from previous timesteps.The location of a point on the interface of domain one (Ω1) is found in terms of the local coordinates of thecorresponding element in Ω2 by treating it as an optimization problem and minimizing residuals with theNewton-Raphson method.15 Explicit interface extrapolation (IEXT) is carried out using the values interpo-lated from data at previous timesteps, with up to third order extrapolation. The mth order extrapolationoperator is given in equation 2 with the coefficients γpm given in table 1.

Em [u]n

=

m∑p=1

γpmun−p (2)

Stability of the extrapolation method has been examined using the one-dimensional unsteady diffusionequation on two overlapping domains with uniform point distribution. The grids overlapped to ensurethe grid points in the two domains coincided. The stability for the extrapolation scheme in the studywas confirmed analytically to be unconditionally stable for first order extrapolation with the backwards-differentiation scheme of first or second order. For higher orders, the stability is dependent upon the meshoverlap size and the number of extrapolation iterations.16

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Table 1. Coefficients for the EXTm schemes, m=1,2,316

γp1 γp2 γp3

p=1 1 2 3

p=2 -1 -3

p=3 1

II.B. Arbitrary Lagrangian-Eulerian Formulation

The Arbitrary Lagrangian Eulerian (ALE) formulation attempts to combine the attractive properties of boththe Lagrangian formulation, where the reference system is attached to ’particles’ in the fluid, and Eulerianformulation, where the reference system is fixed in space. Its main advantage is the ability to assign anarbitrary velocity to the mesh. The velocity assigned to the mesh can be determined by some propertyof the flow, or it can be predetermined and its movement unaffected by the flow. The explanation andderivation of the ALE formulation below follows the general process given by Deville et al.17

The material derivative of a variable, f, given by

Df

Dt=∂f

∂t+ u · ∇f (3)

where u(x, t) is the velocity of the fluid in the Eulerian formulation, describes the time evolution of f attachedto material points. If a mesh is given a velocity of w(x, t) that is not equal to u(x, t), a pseudo-materialderivative can be given for the mesh, as if virtual particles were attached to the moving mesh, which we willcall the ALE derivative

δf

δt=∂f

∂t+ w · ∇f (4)

Note that if w(x, t) = 0 we recover the Eulerian description, or simply the partial time derivative, and ifw(x, t) = u(x, t) we arrive at the material derivative in the Lagrangian description. A relative velocity, c(sometimes called the convective velocity), can be defined with respect to the reference frame of the movingmesh

u = c + w. (5)

The relationship between the material and ALE time derivative can then be given as

Df

Dt=δf

δt+ c · ∇f (6)

To account for mesh movement, the convective terms in the Navier-Stokes equations must be altered toreflect the new relative velocity, c.12 Since density is held constant for incompressible flow, the convectiveterm of the mass conservation equation is zero, and therefore, there is no change in the continuity equation.The momentum conservation equation becomes

δu

δt+ c · ∇u = −∇p +

1

Re∇2u (7)

where c is defined in equation 5. Again, note that if mesh velocity w is equal to zero, the original Navier-Stokes equations 1 are recovered.

II.C. Communicating Updates

Building upon the ALE formulation and interpolation schemes developed previously, additions and modifi-cations were made to allow the two methods to work together. In the original multimesh method, the spatialcoordinates of each interface node are passed to the corresponding processors once at the beginning of asimulation. Since the coordinates are constantly changing in the moving mesh formulation, modificationswere made so that the new coordinates are passed during each timestep. This allows the velocities to becorrectly interpolated/extrapolated during movement of the mesh.

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III. Results: Moving Overlapping Meshes

III.A. Walsh’s Eddies

The first moving mesh benchmark models the convecting solution to Walsh’s eddies18 on two overlappingmeshes, with the inside mesh either rotating or sliding. The initial conditions are given by the streamfunction in equation 8,

ψ(x, y) = −1

5Cos[5x] +

1

4Sin[3x]Sin[4y]− 1

5Sin[5y] + u0y − v0x (8)

giving exact solutions for the velocities of the convecting eddies

u(x, y) = e−25tν(−Cos[5y] + Cos[4y]Sin[3x]) + u0

v(x, y) = e−25tν(−Sin[5x]− 3

4Cos[3x]Sin[4y]) + v0. (9)

This case was used to determine spatial and temporal accuracy and to ensure that the interpola-tion/extrapolation was carried out correctly for moving overlapping meshes without the presence of anysolid objects. Note that although the mesh is moving, there are not any physical structures attached to it,so it may be considered virtual movement since the movement of the mesh should have no affect on the flow.The original mesh configurations (at t=0) are shown in figure 1. The exterior mesh contains a vacancy inthe center that is covered by the interior mesh.

(a) Circular interior mesh (b) Square interior mesh

Figure 1. Two-mesh domains

The second order extrapolation scheme was used, and the angular velocity was fixed at π4 for studying

the spatial and temporal convergence of the method for the rotating cases. The velocity spatial convergenceplots for both the square and circular rotating interior mesh simulations are shown in figure 2 with thenon-rotating case results for comparison. The plots show spectral convergence and trends that correspondwell to the stationary mesh data. The temporal accuracy, shown in figure 3 shows the expected second orderconvergence. Analysis of the pressure accuracy (not shown here) gives comparable results for both spatialand temporal convergence. The results from sliding mesh cases (not shown here) also show spectral accuracyfor spatial discretization, and the expected second order temporal convergence. We see that the convergencerates of the underlying Spectral Element Method solver are maintained with the addition of our movingoverlapping mesh method.

III.B. Oscillating 2D Cylinder in Uniform Flow

Since the earliest research performed by G Magnus19 in 1852, a certain fascination has accompanied theresearch seeking to understand the flow around moving cylinders. The topic of oscillating or vibrating

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1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

2 4 6 8 10 12 14 16 18 20

L2 E

rror

Polynomial Order of Approximation

Interior Mesh

Exterior Mesh

Interior Mesh (St)

Exterior Mesh (St)

(a) Square interior mesh

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

2 4 6 8 10 12 14 16 18 20

L2 E

rror

Polynomial Order of Approximation

Interior Mesh

Exterior Mesh

Interior Mesh (St)

Exterior Mesh (St)

(b) Circular interior mesh

Figure 2. Velocity spatial accuracy for rotating interior mesh simulations using second order time-stepping and EXT.’St’ indicates a stationary mesh case.

1E-08

1E-07

1E-06

1E-05

1E-04

1E-04 1E-03 1E-02

L2 E

rror

Time Step (Δt)

Interior Mesh

Exterior Mesh

Interior Mesh (St)

Exterior Mesh (St)

Δt2

(a) Square interior mesh

1E-08

1E-07

1E-06

1E-05

1E-04 1E-03 1E-02

L2 E

rror

Time Step (Δt)

Interior Mesh

Exterior Mesh

Interior Mesh (St)

Exterior Mesh (St)

Δt2

(b) Circular interior mesh

Figure 3. Velocity temporal accuracy for rotating interior mesh simulations using a polynomial order of approximationof 18. ’St’ indicates a stationary mesh case.

cylinders in cross-flow has been the subject of several experimental and computational studies.20–24 In 1998,Blackburn et al. published results from an oscillating cylinder simulation that utilized a spectral elementscode.25 Their results, along with other published data, established detailed characteristics of the flow whichallowed for comparison and validation of the present work.

For our oscillating cylinder simulations, the domain is assigned a uniform free stream velocity U at theleft boundary with outflow conditions assigned on the opposing side. Symmetry boundary conditions areimposed on the top and bottom of the domain. Spatial units and the amplitude of the cylinder oscillationare non-dimensionalized by the diameter of the cylinder. The horizontal velocity of the cylinder is set tozero, while the vertical component of motion is governed by the equation

y(t) = y0 +Asin(2πf0t), (10)

where A represents the amplitude and f0 represents the frequency of oscillation. The cylinder is initiallyplaced at (0,0), and the domain spans from x = −10 to 50 diameters, and from y = −15 to 15 diameters.

The exterior mesh (see fig. 4) contains a vacancy where the cylinder is placed, and the interior meshwas constructed around the 2D cylinder. During the simulations, the exterior mesh remained stationarywhile the prescribed motion of the interior mesh caused it to move with the equation of oscillation givenby equation 10. The first task involved retrieving the Strouhal number from flow around the stationarycylinder. For a Reynolds number of 200, our simulations produced a Strouhal number of 0.1998 comparedwith the Strouhal number found in Udaykumar et al.24 of 0.198 and the value found in Williamson et al.23

of 0.197. The mean drag of our simulation was 1.372, while Udaykumar et al.24 report a value of 1.38.The drag and lift on an oscillating cylinder in a uniform cross-flow changes with time. However, after a

sufficient startup time, the solution reaches an asymptotic state, indicated by values that repeat periodically

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(a)Inte-riormesh

(b) Exterior mesh

Figure 4. Geometry of the oscillating cylinder case

according to the motion of the cylinder. The results published in Blackburn et al.25 include detailedinformation about the mean drag forces, the peak lift forces, and the mean pressure differences on thecylinder. Using the Strouhal number from stationary mesh simulations, the oscillation frequency of thecylinder is expressed as a ratio of the fixed-cylinder vortex shedding frequency. Table 2 compares theoscillating cylinder results with the results published by Blackburn et al.25 for a frequency ratio of 1.0 andnon-dimensional amplitude 0.25. The label Cl represents the peak coefficient of lift, Cd is the mean coefficientof drag, and Cpb is the mean base pressure coefficient calculated from the pressures at the furthest upstream(p0) and downstream (p180) points on the cylinder surface, Cpb = 1 + 2(p180 − p0)/ρU2.

Blackburn et al. Present Data

Cl 1.776 1.781

Cd 1.414 1.417

−Cpb 1.377 1.377

Table 2. Results for an oscillating cylinder with Re = 500, F = 1 and A = 0.25 in uniform flow compared with resultsfrom Blackburn et al.25

Figure 5. Zoomed-in velocity magnitude plot of a cylinder oscillating with properties: A = 0.25, F = 1, Re = 500

The velocity magnitude plot in figure 5 illustrates the vortices that are shed from an oscillating cylinderwith a non-dimensional amplitude, A = 0.25 and a frequency ratio, F = 1.0, with a Reynolds number,Re = 500. The frequency of vortex shedding, the forces and pressure on the cylinder, and other flowcharacteristics correlate very well with available published data.

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One primary benefit of our overlapping moving mesh method is the ability to model flows that aredifficult or impossible to simulate using traditional methods, such as the flow around solid bodies movingwith extreme ranges of motion. Figure 6 shows the velocity magnitude data from a case with a remarkablylarge amplitude ratio (A=2.5), much larger than any presently found published data. The overlappingmoving mesh method gives accurate results for simulations containing solid bodies, and can simulate flowswith extreme solid body motions that are impossible to model using traditional techniques.

Figure 6. Velocity magnitude representation of the global domain for a cylinder oscillating with properties: A = 2.5,F = 1, Re = 200

III.C. Rotating 3D Cylinders

One of the benefits of an overlapping moving mesh method is the ability to accurately model fluid flowfor cases involving a complex solid body geometry moving in a complex manner. One such situation isa horizontal axis wind turbine (HAWT). A future goal of the present project is to attain accurate dataregarding the blade-wake, and wake-wake interactions in the flow through a rotating HAWT. To ensurethat the method is adequately accurate and robust for these types of three dimensional rotational flows apreliminary test was performed using two cylinders exhibiting a similar motion to a rotating wind turbine.

The interior mesh, whose enlarged cross-section is shown in figure 7(a), was constructed around two 3-dimensional cylinders with diameter D, and the distance between the far ends of the two cylinders is 11D. Thenearest node normal to the surface of the cylinders is approximately 0.03D away. The spatial resolution atthe cylinder walls includes 10 elements in the span-wise direction of each cylinder and 16 elements radiallyaround each cylinder. Note that the solution within each element is computed at several Gauss-Lobattopoints, the number determined by the polynomial order of approximation established before the simulationis executed. The exterior mesh, a selection of which is shown in figure 7(b), contains a vacancy that iscovered by the interior mesh . The global domain extends 30D upstream of the rotating cylinders and 50Ddownstream of the rotating cylinders, and the sides of the domain extend from -30D to 30D. The resolutionof the interior mesh is much finer than that of the exterior mesh allowing greater ability to capture accuratesolutions in the boundary layer.

Figure 8 depicts the vortices in the wake at the -0.05 λ2 isosurface using second order extrapolation. Theflow at the inlet of the domain is set to uniform velocity U, the tip speed ratio (TSR) of the rotating cylindersis 0.55, and the Reynolds number (Re = ρUD/µ) is 200. Realistic wind turbine flows have considerably largerReynolds numbers, and thus in subsequent wind turbine simulations the mesh will be refined to adequatelycapture the small scale fluctuations, especially in the boundary layer region near the blade walls.

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(a) Enlarged cross-section of the inte-rior mesh

(b) Selection of exterior mesh

Figure 7. Geometry of rotating cylinders case

Figure 8. λ2 isosurfaces in the wake of two rotating cylinders with TSR = 0.55 and Re = 200

IV. Conclusion

The validation benchmarks performed have shown good correlation with exact solutions where they areavailable, and with existing experimental and computational data. This new overlapping moving meshmethod used with a Spectral Element Method solver opens doors to accurate flow simulations with movinggeometries. Applications are endless and range from rotary wing aerodynamics, to oscillating pumps andpistons, to stirred reactors. The method will be used in the near future to study highly resolved wakestructures and blade-wake interactions for a rotating wind-turbine rotor. Future development will allow

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the motion to be prescribed according to the forces on the structure, rather than prescribed by the user.Additional development must also be done to enable flow simulations around structures with both movingand stationary parts.

V. Acknowledgements

We would like to acknowledge support for this work given by the National Science Foundation grantCMMI-1250124.

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University Press, 2002.18Walsh, O., “Eddy solutions of the Navier-Stokes equations,” The Navier-Stokes Equations II - Theory and Numerical

Methods, Springer, 1992, pp. 306–309.19Magnus, G., “On the Deflection of a Projectile,” Abhandlungen der Akademie der Wissenschaften, Berlin, Germany,

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